Specific sequences of observations can lead to ingenuous ways to correct for the telluric absorption. The telluric lines are stable down to 10 to 20 m.s−1 in the Earth referential (Figueira et al., 2010a), but when a star is observed from Earth, the telluric lines follow the same radial velocity variation as the barycentric radial velocity of the Earth. In this case, the spectra are imprinted with telluric lines whose radial velocity can vary from -30 to 30 km.s−1. This particularity allows one to recognize and identify the telluric lines against stellar lines.
2.4 Data-driven methods 25
2.4.1 Wobble
A recent paper published by Bedell et al. (2019) presents a novel method to correct the telluric absorption and derive radial velocity measurements at the same time. Wobble is a data-driven method which does not rely on a stellar template nor an atmospheric transmission to retrieve the stellar spectrum free of telluric lines. The method uses an input data set of 1D wavelength calibrated spectra from a stabilized high-resolution spectrograph with no additional gas cell imprinted in the science spectrum (e.g. iodine cell). To ensure that the telluric correction is efficient, the input spectra should cover different observing epochs during the year, allowing spectra to be registered at different barycentric radial velocities of the Earth.
Wobble assumes that each spectrum can be represented as the product of a telluric spectrum and a stellar spectrum. In the reference frame of the observatory, the telluric spectrum is supposed to be fixed which means that the telluric lines are found at the same wavelengths, but the stellar spectrum varies in position through the RV imprinted by Earth’s rotation around the Sun. In terms of shape, the stellar spectrum has a fixed shape which means for example that the stellar lines have the same absorption, while the telluric spectrum has a shape which can vary over time. The contrast of the telluric lines is a function of the airmass measured at the time of the observation. The first guess for the stellar spectrum is provided by computing the median combined spectrum of the input spectra corrected for the barycentric radial velocity of the Earth (BERV).
The telluric spectrum is initialized by computing the median combined spectrum of the input data not corrected for the BERV. Then, the model is optimized with the maximum likelihood estimation method. Both the stellar and telluric spectra are subject to a combination of L1 and L2 regularizations to avoid over-fitting. The use of regularization assumes that the logarithm of the flux of both the telluric and stellar spectra are close to zero. Applying regularization terms will guarantee that the optimization of the free parameters will push both spectra close to zero in case there is no other evidence (Bedell et al., 2019).
After optimization, Wobble derives a radial velocity measurement for each order of the echelle spectrum and delivers a decomposition of the original spectral order into a telluric spectrum and a stellar spectrum. Figure 2.5 presents one order of the HARPS spectrum of 51 Pegasi with the two resulting spectra. The Hα absorption feature is identified as part of the stellar spectrum and even small telluric lines are identified as part of the telluric spectrum. Wobble is a promising method to derive radial velocity and correct for the telluric contamination for datasets which verify the initial assumptions of the method such as the large BERV coverage of the spectra. The authors have plans to extend Wobble to be able to handle spectra from several spectrographs.
26 Chapter 2 Telluric contamination and correction
6555 6560 6565 6570 6575 Wavelength (˚A)
0.2 0.4 0.6 0.8 1.0
NormalizedFlux
Input spectrum Telluric spectrum Stellar spectrum
Fig. 2.5.: The HARPS input spectrum of 51 Peg is plotted in black. The modelled telluric spectrum is plotted in red. The modelled stellar spectrum, thus a telluric free spectrum is plotted in purple. Figure made with the code and dataset from Wobble GitHub repository12.
2.4 Data-driven methods 27
2.4.2 PCA-based telluric correction
The telluric correction published by Artigau et al. (2014) is based on Principal Component Analysis (PCA). Principal component analysis is used as a dimensionality reduction technique, aiming at describing a high-dimension dataset with a reduced number of dimensions. The original dataset is projected on a new basis of vectors, where the variables are linearly uncorrelated and called principal components. In this new basis, the first principal component describes the largest possible variance, and the second principal component is orthogonal to the first and describes the second largest variance.
The following components are built with the same principles. In the case of the telluric correction, the idea is to build a basis of components reflecting the absorption of the Earth’s atmosphere.
In general, the transmission of the Earth’s atmosphere is the ratio between the incident intensity (I0) at the top of the atmosphere and the transmitted intensity (I) at the bottom of the atmosphere:
T = I I0
The opacity of the atmosphere O is then defined as the inverse of the transmission, and the atmospheric absorbance A as:
A = −log(T )
The atmospheric absorption spectrum is considered a finite sum of absorbances by different atmospheric molecules such as H20 and O2. The telluric absorption spectrum can be decomposed as a linear combination of absorbance components. An absorption by one molecule such as water vapour can be found in several absorbance components.
Artigau et al. (2014) identify the absorbance components by running a principal com-ponent analysis on a dataset of telluric standard star spectra observed under various weather conditions and airmasses. As an example, the telluric correction of the τ Ceti dataset used about 200 observations of 30 different telluric standard stars to build the database of individual absorbances. The dataset of science spectra which need the telluric correction is composed of several spectra of the same target observed at different epochs so that the position of the telluric lines relative to the stellar lines is varying with the barycentric velocity of the Earth. Using these data, the telluric correction takes place in two steps. The first step is to fit the science spectra with a linear combination of the first absorbance components of the database and a star spectrum initialized to 1 at all wavelengths. The result of the fit is an approximation to the telluric absorption present in the spectra. This approximation is subtracted from all the science spectra.
The science spectra are then corrected for the BERV so that the stellar lines are all aligned. Then the spectra are median combined to create a new estimate of the stellar
28 Chapter 2 Telluric contamination and correction
spectrum. Finally, a second fit of the science spectra to the absorbance components and the new stellar spectrum is performed. This fit results in a better correction of the telluric absorption and its removal from the science spectra. In practice, Artigau et al.
(2014) find that two iterations are enough to obtain a telluric correction at the observed noise level.
While the PCA-based telluric correction method uses telluric standard star spectra, these observations can be done early in the night, during bad weather or poor seeing conditions.
The loss of telescope time is greatly reduced which is a large advantage compared to the historical standard star method. The τ Ceti dataset has been used to measure the radial velocity of the star: Artigau et al. (2014) shows that the scatter of the radial velocity measurements decreases when the spectra are telluric corrected with their PCA method. This telluric correction method is promising for the near-infrared domain where the telluric contamination is one of most limiting factor to radial velocity precision.